> On one occasion I had two days of travel booked, and the very first train was an hour late, which led to a missed connection, and that was it - there was no way I could make my next train, and I lost some hundreds of euros of booked tickets and accommodation

This is one thing that Amtrak does well. I've read many accounts of people missing a connection and being put up in a nice hotel by Amtrak with meal vouchers and a shuttle to/from the station. Still it's not a good way to travel if you have a deadline, and it often costs an order of magnitude more than flying for long distances unless you're willing to spend multiple day/night cycles in coach.